In 2 hours, the clock will gain 10 minutes (5 minutes/hour * 2 hours). So, when the clock shows 12:00 PM, the actual time will be 12:00 PM - 10 minutes = 11:50 AM.

At 3:15, the hour hand is at 97.5 degrees (3*30 + 15*0.5) and the minute hand is at 90 degrees. The angle between them is |97.5 - 90| = 7.5 degrees.

At 8:30, the hour hand is at 255 degrees (8 * 30 + 30 * 0.5) and the minute hand is at 180 degrees (30 * 6). The angle between them is |255 - 180| = 75 degrees.

The hands of a clock are at right angles at approximately 2:45.

The hands of a clock are at right angles at 3:15 and 3:45. The first occurrence is at 3:15.

The hands of the clock are at right angles at approximately 4:20.

The hour hand moves 30 degrees in 1 hour (360 degrees / 12 hours).

The minute hand moves 6 degrees per minute. In 15 minutes, it moves 15 * 6 = 90 degrees.

The hands of a clock overlap 11 times in a 12-hour period.

If the clock is 10 minutes fast, it will show 2:10 PM when the actual time is 2:00 PM.

If the actual time is 2:00 PM and the clock is 10 minutes fast, it will show 2:10 PM.

If the clock is 10 minutes fast, when it shows 5:50, the actual time is 5:50 - 10 minutes = 5:40.

In 5 hours, the clock will lose 10 minutes (2 minutes/hour * 5 hours). So, it will show 12:00 - 10 minutes = 11:50.

At 10:10, the hour hand is at 85 degrees (10*30 + 10*0.5) and the minute hand is at 60 degrees. The angle between them is |85 - 60| = 25 degrees.

At 12:45, the hour hand is at 337.5 degrees (12 * 30 + 45 * 0.5) and the minute hand is at 270 degrees (45 * 6). The angle between them is |337.5 - 270| = 67.5 degrees.

At 3:15, the hour hand is at 97.5 degrees (3 hours * 30 + 15 minutes * 0.5) and the minute hand is at 90 degrees (15 minutes * 6). The angle between them is |97.5 - 90| = 7.5 degrees.

At 4:20, the hour hand is at 130 degrees (4*30 + 20*0.5) and the minute hand is at 120 degrees (20*6). The angle between them is |130 - 120| = 10 degrees.

At 6:00, the hour hand is at 180 degrees and the minute hand is at 0 degrees. The angle between them is |180 - 0| = 180 degrees.

Adding 3 hours and 45 minutes to 6:00 PM gives 9:45 PM.

Adding 3 hours and 45 minutes to 6:00 PM gives 9:45 PM.

At 9:15, the hour hand is at 97.5 degrees (9*30 + 15*0.5) and the minute hand is at 90 degrees. The angle between them is |97.5 - 90| = 7.5 degrees.

In 15 minutes, the minute hand moves (360 degrees / 60 minutes) * 15 minutes = 90 degrees.

At 1:50, the hour hand is at 95 degrees (1 hour * 30 degrees + 50 minutes * 0.5 degrees) and the minute hand is at 300 degrees (50 minutes * 6 degrees). The angle between them is |95 - 300| = 205 degrees.

At 2:30, the hour hand is at 75 degrees (2 hours * 30 degrees + 30 minutes * 0.5 degrees) and the minute hand is at 180 degrees (30 minutes * 6 degrees). The angle between them is |75 - 180| = 105 degrees.

At 12:30, the hour hand is at 165 degrees (12*30 + 30*0.5) and the minute hand is at 180 degrees. The angle between them is |165 - 180| = 15 degrees.

At 12:45, the hour hand is at 337.5 degrees (12 hours * 30 degrees + 45 minutes * 0.5 degrees) and the minute hand is at 270 degrees (45 minutes * 6 degrees). The angle between them is |337.5 - 270| = 67.5 degrees.

At 5:25, the hour hand is at 162.5 degrees (5 hours * 30 degrees + 25 minutes * 0.5 degrees) and the minute hand is at 150 degrees (25 minutes * 6 degrees). The angle between them is |162.5 - 150| = 12.5 degrees.

At 12:30, the hour hand is at 165 degrees (12*30 + 30*0.5) and the minute hand is at 180 degrees (30*6). The angle between them is |165 - 180| = 15 degrees.

The time difference is 4:15 PM - 2:30 PM = 1 hour 45 minutes.

At 6:00, the hour hand is at 180 degrees and the minute hand is at 0 degrees. The angle between them is 180 degrees.